Characterizing a Data Distribution Unit 6A Characterizing a Data Distribution

DISTRIBUTION The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values. However, much of the time we are less interested in the complete distribution than in a few descriptive terms that summarize it.

AVERAGE There a three different terms that characterize the center of a data distribution. Any of these can be called the “average.” Mean Median Mode

MEAN The mean is defined as: The mean is what we most commonly refer to as the average value.

EXAMPLE The given values are the numbers of Dutchess County car crashes for each month in a recent year. Find the mean. 27 8 17 11 15 25 16 14 14 14 13 18

MEDIAN The median is the middle value in a sorted data set (or halfway between the two middle values if the number of values is even).

(even number of values) 6.72 3.46 3.60 6.44 3.46 3.60 6.44 6.72 no exact middle -- shared by two numbers 3.60 + 6.44 2 (even number of values) MEDIAN is 5.02 6.72 3.46 3.60 6.44 26.70 3.46 3.60 6.44 6.72 26.70 (in order - odd number of values) exact middle MEDIAN is 6.44

EXAMPLE The given values are the numbers of Dutchess County car crashes for each month in a recent year. Find the median. 27 8 17 11 15 25 16 14 14 14 13 18

MODE The mode is the most common value (or group of values) in a distribution. When two values occur with the same greatest frequency, each one is a mode and the data set is said to be bimodal. When more than two values occur with the same greatest frequency, each is a mode and the data set is said to be multimodal.

EXAMPLE a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Mode is 5 Bimodal - 2 and 6 No Mode

EXAMPLE The given values are the numbers of Dutchess County car crashes for each month in a recent year. Find the mode. 27 8 17 11 15 25 16 14 14 14 13 18

FINDING THE MEAN AND MEDIAN ON THE TI-83/84 Press STAT; select 1:Edit…. Enter your data values in L1. (You may enter the values in any of the lists.) Press 2ND, MODE (for QUIT). Press STAT; arrow over to CALC. Select 1:1-Var Stats. Enter L1 by pressing 2ND, 1. Press ENTER. The mean is the number after x=. Scroll down using the down arrow key to find the median. It is on the line Med=.

FINDING THE MODE ON THE TI-83/84 The TI-83/84 will NOT calculate the mode of a data set. However, the data in a list can be easily sorted to help in finding the mode. To sort L1 in ascending (lowest to highest) order: STAT, 2:SortA, L1, ), and ENTER. To sort in descending (highest to lowest) order, use 3:SortD.

OUTLIER An outlier is a data value that is much higher or lower than almost all other values. Outliers usually have a drastic affect on the mean. However, outliers do not affect the median or the mode.

SYMMETRIC A distribution is symmetric if its left half is a mirror image of its right half. Mode = Mean = Median NOTE: The mean, median, and mode are equal. SYMMETRIC

LEFT-SKEWED A distribution is left-skewed if its values are more spread out on the left side. Another term for left-skewed is negatively-skewed. NOTE: The mean is on the left; the mode on the right; and the median in between the mean and mode. Mean Mode Median LEFT-SKEWED (negatively)

RIGHT-SKEWED A distribution is right-skewed if its values are more spread out on the right side. Another term for right-skewed is positively-skewed. Median Mean Mode NOTE: The mode is on the left; the mean on the right; and the median in between the mean and mode. RIGHT-SKEWED (positively)

VARIATION Variation describes how widely data values are spread out about the center of a distribution.